Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0902.4153 · Physical Review B

Location and properties of the multicritical point in the Gaussian and ± J Ising spin glasses

S. L. A. de Queiroz

Published May 7, 2009• 27 citations
arXivHTMLPDF

Abstract

We use transfer-matrix and finite-size scaling methods to investigate the location and properties of the multicritical point of two-dimensional Ising spin glasses on square, triangular, and honeycomb lattices with both binary and Gaussian disorder distributions. For square and triangular lattices with binary disorder, the estimated position of the multicritical point is in numerical agreement with recent conjectures regarding its exact location. For the remaining four cases, our results indicate disagreement with the respective versions of the conjecture, though by very small amounts, never exceeding 0.2%. Our results for (i) the correlation-length exponent $\ensuremath{\nu}$ governing the ferroparamagnetic transition, (ii) the critical domain-wall energy amplitude $\ensuremath{\eta}$, (iii) the conformal anomaly $c$, (iv) the finite-size susceptibility exponent $\ensuremath{\gamma}/\ensuremath{\nu}$, and (v) the set of multifractal exponents ${{\ensuremath{\eta}}_{k}}$ associated to the moments of the probability distribution of spin-spin correlation functions at the multicritical point are consistent with universality as regards lattice structure and disorder distribution and in good agreement with existing estimates.

DOI
10.1103/physrevb.79.174408

Related papers

Multicritical point of Ising spin glasses on triangular and honeycomb lattices

S. L. A. de Queiroz

The behavior of two-dimensional Ising spin glasses at the multicritical point on triangular and honeycomb lattices is investigated with the help of finite-size scaling and conformal-invariance concepts. We use transfer-matrix methods on long strips to calculate domain-wall energies, uniform susceptibilities, and spin-spin correlation functions. Accurate estimates are provided for the location of the multicritical point on both lattices, which lend strong support to a conjecture recently advanced by Takeda, Sasamoto, and Nishimori. Correlation functions are shown to obey rather strict conformal-invariance requirements, once suitable adaptations are made to account for geometric aspects of the transfer-matrix description of triangular and honeycomb lattices. The universality class of critical behavior upon crossing the ferro-para-magnetic phase boundary is probed, with the following estimates for the associated critical indices: $\ensuremath{\nu}=1.49(2)$, $\ensuremath{\gamma}=2.71(4)$, ${\ensuremath{\eta}}_{1}=0.183(3)$, which are distinctly different from the percolation values.

arXiv:cond-mat/0510816•Physical Review B•Feb 8, 2006•42 citations

Properties of the multicritical point of ±J Ising spin glasses on the square lattice

Jean C. Lessa, S. L. A. de Queiroz

We use numerical transfer-matrix methods to investigate properties of the multicritical point of binary Ising spin glasses on a square lattice, whose location we assume to be given exactly by a conjecture advanced by Nishimori and Nemoto. We calculate the two largest Lyapunov exponents, as well as linear and nonlinear zero-field uniform susceptibilities, on strip of widths $4\ensuremath{\leqslant}L\ensuremath{\leqslant}16$ sites, from which we estimate the conformal anomaly $c$, the decay-of-correlations exponent $\ensuremath{\eta}$, and the linear and nonlinear susceptibility exponents $\ensuremath{\gamma}∕\ensuremath{\nu}$ and ${\ensuremath{\gamma}}^{\mathrm{nl}}∕\ensuremath{\nu}$, with the help of finite-size scaling and conformal invariance concepts. Our results are $c=0.46(1)$; $0.187\ensuremath{\lesssim}\ensuremath{\eta}\ensuremath{\lesssim}0.196$; $\ensuremath{\gamma}∕\ensuremath{\nu}=1.797(5)$; ${\ensuremath{\gamma}}^{\mathrm{nl}}∕\ensuremath{\nu}=5.59(2)$. A direct evaluation of correlation functions on the strip geometry, and of the statistics of the zeroth moment of the associated probability distribution, gives $\ensuremath{\eta}=0.194(1)$, consistent with the calculation via Lyapunov exponents. Overall, these values tend to be inconsistent with the universality class of percolation, though by small amounts. The scaling relation ${\ensuremath{\gamma}}^{\mathrm{nl}}∕\ensuremath{\nu}=2\ensuremath{\gamma}∕\ensuremath{\nu}+d$ (with space dimensionality $d=2$) is obeyed to rather good accuracy, thus showing no evidence of multiscaling behavior of the susceptibilities.

arXiv:cond-mat/0605659•Physical Review B•Oct 24, 2006•6 citations

Correlation-function distributions at the Nishimori point of two-dimensional Ising spin glasses

S. L. A. de Queiroz, Robin B. Stinchcombe

The multicritical behavior at the Nishimori point of two-dimensional Ising spin glasses is investigated by using numerical transfer-matrix methods to calculate probability distributions $P(C)$ and associated moments of spin-spin correlation-functions C on strips. The angular dependence of the shape of correlation function distributions $P(C)$ provides a stringent test of how well they obey predictions of conformal invariance; and an even symmetry of $(1\ensuremath{-}C)P(C)$ reflects the consequences of the Ising spin-glass gauge (Nishimori) symmetry. We show that conformal invariance is obeyed in its strictest form, and the associated scaling of the moments of the distribution is examined, in order to assess the validity of a recent conjecture on the exact localization of the Nishimori point. Power-law divergences of $P(C)$ are observed near $C=1$ and $C=0,$ in partial accord with a simple scaling scheme which preserves the gauge symmetry.

arXiv:cond-mat/0303128•Physical review. B, Condensed matter•Oct 8, 2003•16 citations

Multicritical Points of Potts Spin Glasses on the Triangular Lattice

Masayuki Ohzeki

We predict the locations of several multicritical points of the Potts spin glass model on the triangular lattice. In particular, continuous multicritical lines, which consist of multicritical points, are obtained for two types of two-state Potts (i.e., Ising) spin glasses with two- and three-body interactions on the triangular lattice. These results provide us with numerous examples to further verify the validity of the conjecture, which has succeeded in deriving highly precise locations of multicritical points for several spin glass models. The technique, called the direct triangular duality, a variant of the ordinary duality transformation, directly relates the triangular lattice with its dual triangular lattice in conjunction with the replica method.

arXiv:0706.2052•Journal of the Physical Society of Japan•Oct 24, 2007•3 citations

Finite-size scaling study of the two-dimensional Ising spin glass

David A. Huse, I. Morgenstern

We have calculated finite-size correlation lengths for strips of the square-lattice Ising spin glass using the transfer-matrix method. In order to minimize sampling errors we study strip lengths up to 5\ifmmode\times\else\texttimes\fi{}${10}^{6}$ lattice spacings. A phenomenological renormalization-group analysis indicates that there are strong corrections to simple power-law scaling near the zero-temperature critical point, as is to be expected near a lower critical dimension. We examine models with Gaussian, exp(-${J}^{2}$/2), and exponential, exp(-\ensuremath{\Vert}J\ensuremath{\Vert}), distributions of couplings; the Gaussian distribution shows stronger finite-size corrections. The correlation-length exponent is estimated to be \ensuremath{\nu}=4.2\ifmmode\pm\else\textpm\fi{}0.5, although we do not want to rule out the possibility that \ensuremath{\nu} is significantly larger than this.

Physical review. B, Condensed matter•Sep 1, 1985•40 citations

Strong disorder fixed points in the two-dimensional random-bond Ising model

Marco Picco, A. Honecker, Pierre Pujol

The random-bond Ising model on the square lattice has several disordered critical points, depending on the probability distribution of the bonds. There are a finite-temperature multicritical point, called the Nishimori point, and a zero-temperature fixed point, for both a binary distribution where the coupling constants take the values ± J and a Gaussian disorder distribution. Inclusion of dilution in the ± J distribution ( J = 0 for some bonds) gives rise to another zero-temperature fixed point which can be identified with percolation in the non-frustrated case ( J ≥ 0). We study these fixed points using numerical (transfer matrix) methods. We determine the location, critical exponents, and central charge of the different fixed points and study the spin–spin correlation functions. Our main findings are the following. (1) We confirm that the Nishimori point is universal with respect to the type of disorder, i.e. we obtain the same central charge and critical exponents for the ± J and Gaussian distributions of disorder. (2) The Nishimori point, the zero-temperature fixed point for the ± J and Gaussian distributions of disorder, and the percolation point in the diluted case all belong to mutually distinct universality classes. (3) The paramagnetic phase is re-entrant below the Nishimori point, i.e. the zero-temperature fixed points are not located exactly below the Nishimori point, either for the ± J distribution, or for the Gaussian distribution.

arXiv:cond-mat/0606312•Journal of Statistical Mechanics Theory and Experiment•Sep 13, 2006•50 citations
We use transfer-matrix and finite-size scaling methods to investigate the location and properties of the multicritical point of two-dimensional Ising spin glasses on square, triangular, and honeycomb lattices with both binary and Gaussian disorder distributions. For square and triangular lattices with binary disorder, the estimated position of the multicritical point is in numerical agreement with recent conjectures regarding its exact location. For the remaining four cases, our results indicate disagreement with the respective versions of the conjecture, though by very small amounts, never exceeding 0.2%. Our results for (i) the correlation-length exponent \ensuremathν\ensuremath{\nu}\ensuremathν governing the ferroparamagnetic transition, (ii) the critical domain-wall energy amplitude \ensuremathη\ensuremath{\eta}\ensuremathη, (iii) the conformal anomaly ccc, (iv) the finite-size susceptibility exponent \ensuremathγ/\ensuremathν\ensuremath{\gamma}/\ensuremath{\nu}\ensuremathγ/\ensuremathν, and (v) the set of multifractal exponents \ensuremathηk{{\ensuremath{\eta}}_{k}}\ensuremathηk​ associated to the moments of the probability distribution of spin-spin correlation functions at the multicritical point are consistent with universality as regards lattice structure and disorder distribution and in good agreement with existing estimates.
The behavior of two-dimensional Ising spin glasses at the multicritical point on triangular and honeycomb lattices is investigated with the help of finite-size scaling and conformal-invariance concepts. We use transfer-matrix methods on long strips to calculate domain-wall energies, uniform susceptibilities, and spin-spin correlation functions. Accurate estimates are provided for the location of the multicritical point on both lattices, which lend strong support to a conjecture recently advanced by Takeda, Sasamoto, and Nishimori. Correlation functions are shown to obey rather strict conformal-invariance requirements, once suitable adaptations are made to account for geometric aspects of the transfer-matrix description of triangular and honeycomb lattices. The universality class of critical behavior upon crossing the ferro-para-magnetic phase boundary is probed, with the following estimates for the associated critical indices: $\ensuremath{\nu}=1.49(2)$, $\ensuremath{\gamma}=2.71(4)$, ${\ensuremath{\eta}}_{1}=0.183(3)$, which are distinctly different from the percolation values.
We use numerical transfer-matrix methods to investigate properties of the multicritical point of binary Ising spin glasses on a square lattice, whose location we assume to be given exactly by a conjecture advanced by Nishimori and Nemoto. We calculate the two largest Lyapunov exponents, as well as linear and nonlinear zero-field uniform susceptibilities, on strip of widths $4\ensuremath{\leqslant}L\ensuremath{\leqslant}16$ sites, from which we estimate the conformal anomaly ccc, the decay-of-correlations exponent $\ensuremath{\eta}$, and the linear and nonlinear susceptibility exponents $\ensuremath{\gamma}∕\ensuremath{\nu}$ and ${\ensuremath{\gamma}}^{\mathrm{nl}}∕\ensuremath{\nu}$, with the help of finite-size scaling and conformal invariance concepts. Our results are c=0.46(1)c=0.46(1)c=0.46(1); $0.187\ensuremath{\lesssim}\ensuremath{\eta}\ensuremath{\lesssim}0.196$; $\ensuremath{\gamma}∕\ensuremath{\nu}=1.797(5)$; ${\ensuremath{\gamma}}^{\mathrm{nl}}∕\ensuremath{\nu}=5.59(2)$. A direct evaluation of correlation functions on the strip geometry, and of the statistics of the zeroth moment of the associated probability distribution, gives $\ensuremath{\eta}=0.194(1)$, consistent with the calculation via Lyapunov exponents. Overall, these values tend to be inconsistent with the universality class of percolation, though by small amounts. The scaling relation ${\ensuremath{\gamma}}^{\mathrm{nl}}∕\ensuremath{\nu}=2\ensuremath{\gamma}∕\ensuremath{\nu}+d$ (with space dimensionality d=2d=2d=2) is obeyed to rather good accuracy, thus showing no evidence of multiscaling behavior of the susceptibilities.
The multicritical behavior at the Nishimori point of two-dimensional Ising spin glasses is investigated by using numerical transfer-matrix methods to calculate probability distributions P(C)P(C)P(C) and associated moments of spin-spin correlation-functions C on strips. The angular dependence of the shape of correlation function distributions P(C)P(C)P(C) provides a stringent test of how well they obey predictions of conformal invariance; and an even symmetry of $(1\ensuremath{-}C)P(C)$ reflects the consequences of the Ising spin-glass gauge (Nishimori) symmetry. We show that conformal invariance is obeyed in its strictest form, and the associated scaling of the moments of the distribution is examined, in order to assess the validity of a recent conjecture on the exact localization of the Nishimori point. Power-law divergences of P(C)P(C)P(C) are observed near C=1C=1C=1 and C=0,C=0,C=0, in partial accord with a simple scaling scheme which preserves the gauge symmetry.
We predict the locations of several multicritical points of the Potts spin glass model on the triangular lattice. In particular, continuous multicritical lines, which consist of multicritical points, are obtained for two types of two-state Potts (i.e., Ising) spin glasses with two- and three-body interactions on the triangular lattice. These results provide us with numerous examples to further verify the validity of the conjecture, which has succeeded in deriving highly precise locations of multicritical points for several spin glass models. The technique, called the direct triangular duality, a variant of the ordinary duality transformation, directly relates the triangular lattice with its dual triangular lattice in conjunction with the replica method.
We have calculated finite-size correlation lengths for strips of the square-lattice Ising spin glass using the transfer-matrix method. In order to minimize sampling errors we study strip lengths up to 5\ifmmode\times\else\texttimes\fi{}106{10}^{6}106 lattice spacings. A phenomenological renormalization-group analysis indicates that there are strong corrections to simple power-law scaling near the zero-temperature critical point, as is to be expected near a lower critical dimension. We examine models with Gaussian, exp(-J2{J}^{2}J2/2), and exponential, exp(-\ensuremath{\Vert}J\ensuremath{\Vert}), distributions of couplings; the Gaussian distribution shows stronger finite-size corrections. The correlation-length exponent is estimated to be \ensuremath{\nu}=4.2\ifmmode\pm\else\textpm\fi{}0.5, although we do not want to rule out the possibility that \ensuremath{\nu} is significantly larger than this.
The random-bond Ising model on the square lattice has several disordered critical points, depending on the probability distribution of the bonds. There are a finite-temperature multicritical point, called the Nishimori point, and a zero-temperature fixed point, for both a binary distribution where the coupling constants take the values ± J and a Gaussian disorder distribution. Inclusion of dilution in the ± J distribution ( J = 0 for some bonds) gives rise to another zero-temperature fixed point which can be identified with percolation in the non-frustrated case ( J ≥ 0). We study these fixed points using numerical (transfer matrix) methods. We determine the location, critical exponents, and central charge of the different fixed points and study the spin–spin correlation functions. Our main findings are the following. (1) We confirm that the Nishimori point is universal with respect to the type of disorder, i.e. we obtain the same central charge and critical exponents for the ± J and Gaussian distributions of disorder. (2) The Nishimori point, the zero-temperature fixed point for the ± J and Gaussian distributions of disorder, and the percolation point in the diluted case all belong to mutually distinct universality classes. (3) The paramagnetic phase is re-entrant below the Nishimori point, i.e. the zero-temperature fixed points are not located exactly below the Nishimori point, either for the ± J distribution, or for the Gaussian distribution.